Best Known (54−14, 54, s)-Nets in Base 16
(54−14, 54, 18725)-Net over F16 — Constructive and digital
Digital (40, 54, 18725)-net over F16, using
- net defined by OOA [i] based on linear OOA(1654, 18725, F16, 14, 14) (dual of [(18725, 14), 262096, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(1654, 131075, F16, 14) (dual of [131075, 131021, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(1654, 131076, F16, 14) (dual of [131076, 131022, 15]-code), using
- trace code [i] based on linear OA(25627, 65538, F256, 14) (dual of [65538, 65511, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(25627, 65536, F256, 14) (dual of [65536, 65509, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(25625, 65536, F256, 13) (dual of [65536, 65511, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- trace code [i] based on linear OA(25627, 65538, F256, 14) (dual of [65538, 65511, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(1654, 131076, F16, 14) (dual of [131076, 131022, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(1654, 131075, F16, 14) (dual of [131075, 131021, 15]-code), using
(54−14, 54, 73355)-Net over F16 — Digital
Digital (40, 54, 73355)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1654, 73355, F16, 14) (dual of [73355, 73301, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(1654, 131072, F16, 14) (dual of [131072, 131018, 15]-code), using
- trace code [i] based on linear OA(25627, 65536, F256, 14) (dual of [65536, 65509, 15]-code), using
- an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- trace code [i] based on linear OA(25627, 65536, F256, 14) (dual of [65536, 65509, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(1654, 131072, F16, 14) (dual of [131072, 131018, 15]-code), using
(54−14, 54, large)-Net in Base 16 — Upper bound on s
There is no (40, 54, large)-net in base 16, because
- 12 times m-reduction [i] would yield (40, 42, large)-net in base 16, but